A stratification of Hilbert schemes by initial ideals and applications

We show that the set of the homogeneous saturated ideals with given initial ideal in a fixed term-ordering is locally closed in the Hilbert scheme, and that it is affine if the initial ideal is saturated. Then, Hilbert schemes can be stratified using these subschemes. We investigate the behaviour of this stratification with respect to some properties of the closed points. As application, we describe the singular locus of the component of Hilb(P4)(4z+1) containing the ACM curves of degree 4.